is extensive and we
mention [1]}, [2]} and
[3]} for an
introduction and detailed bibliographies.
Restricting to large \(H^{1}\)  (energy class) data,
Problem (REF ) is well posed for
large data below the critical value
\(p<p_{0}(n)\)  where
\(p_{0}(n)=\frac{n+2}{n-2}\)  for \(n>2\)  and \(p_{0}(n)=+\infty \) 
if \(n=1,2\)  ([4]}).
The problem is well posed also in the critical case
\(p=p_{0}(n)\)  as proved in a series of important papers
([5]},
[6]},
[7]},
[8]},
[9]},
[10]},
[11]}).
Well or ill posedness
in the supercritical case \(p>p_{0}(n)\) 
has been for many years a completely open problem.
A recent breaktrhough was obtained in
[12]}, where finite time blow up
was established for a class of large, radially symmetric,
localized initial data and suitable ranges of \((n,p)\) .
